It completely depends on what you want to see. The west has more natural beauty, in my opinion, and you are close to things like the Cliffs of Moher, the Connemara area, the Dingle peninsula and the Killarney National Park area. On the Dublin side, you are nearer to Belfast, Newgrange, the Wicklow Mountains, Kilkenny and Cashel. You can't really go wrong with whatever you choose. If you were driving, I would definitely say it would be better to land in Shannon just because it is easier to get used to driving on the other side of the road there because of the lack of traffic.

Also remember that you don't have to fly out of the same airport you flew into, if that works better for your plans. To find these flights (for instance, into Shannon and out of Dublin), use the "multi city" option on Kayak, Expedia, etc. Don't try to buy two one-way tickets - that's much more expensive.